HSC Science (General) 12th Board ExamMaharashtra State Board
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Solution - Find the equation of a curve passing through the point (0, 2), given that the sum of the coordinates of any point on the curve exceeds the slope of the tangent to the curve at that point by 5 - HSC Science (General) 12th Board Exam - Mathematics and Statistics

Question

Find the equation of a curve passing through the point (0, 2), given that the sum of the coordinates of any point on the curve exceeds the slope of the tangent to the curve at that point by 5

Solution

Since, `dy/dx` represents the slope of tangent to a given curve at a point (x, y), the given equation is

`dy/dx+5 = x+y`

`therefore dy/dx-y=(x-5)`

The given equation is of the form `dy/dx+Py=Q`

where, `P=-1 " and " Q=(x-5)`

`therefore I.F. = e^(intPdx)=e^(int-1dx)=e^-x`

Solution of the given equation is

`y(I.F.) = intQ(I.F.)dx+c`

`therefore ye^-x=int(x-5)e^-xdx+c`

`=intxe^-xdx - 5int e^-xdx+c`

`=x inte^-xdx-int[d/dx(x) inte^-xdx]dx-5e^-x/-1+c`

`-xe^-x-inte^-x/-1dx+5e^-x + c`

`-xe^-x+inte^-xdx+5e^-x + c`

`-xe^-x-e^-x+5e^-x + c`

`therefore ye^-x = -xe^-x+4e^-x+c`

`therefore y=-x+4+ce^x`

`therefore x+y-4=ce^x` is the general solution

Since the curve is passing through the point (0,2)

`therefore x = 0, y = 2`

`therefore 0+2-4=ce^0`

`therefore c=-2`

`therefore x+y-4=-2e^x`

`therefore y=4-x-2e^x` is the required equation of the curve.

 

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APPEARS IN

 2015-2016 (July) (with solutions)
Question 6.2.1 | 4 marks

Video TutorialsVIEW ALL [1]

Reference Material

Solution for question: Find the equation of a curve passing through the point (0, 2), given that the sum of the coordinates of any point on the curve exceeds the slope of the tangent to the curve at that point by 5 concept: Area of the Region Bounded by a Curve and a Line. For the courses HSC Science (General) , HSC Science (Computer Science), HSC Science (Electronics), HSC Arts
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